NP-Completeness NP-Completeness Graphs 5/7/ :49 PM x x x x x x x

Slides:



Advertisements
Similar presentations
Complexity class NP Is the class of languages that can be verified by a polynomial-time algorithm. L = { x in {0,1}* | there exists a certificate y with.
Advertisements

Computability and Complexity 13-1 Computability and Complexity Andrei Bulatov The Class NP.
Graphs 4/16/2017 8:41 PM NP-Completeness.
NP-Complete Problems Reading Material: Chapter 10 Sections 1, 2, 3, and 4 only.
The Theory of NP-Completeness
NP-complete and NP-hard problems
Analysis of Algorithms CS 477/677
NP-Completeness NP-Completeness Graphs 4/17/2017 4:10 AM x x x x x x x
Chapter 11: Limitations of Algorithmic Power
Toward NP-Completeness: Introduction Almost all the algorithms we studies so far were bounded by some polynomial in the size of the input, so we call them.
1 CSE 417: Algorithms and Computational Complexity Winter 2001 Lecture 22 Instructor: Paul Beame.
February 20, 2015CS21 Lecture 191 CS21 Decidability and Tractability Lecture 19 February 20, 2015.
NP-Completeness x 1 x 3 x 2 x 1 x 4 x 3 x 2 x
TECH Computer Science NP-Complete Problems Problems  Abstract Problems  Decision Problem, Optimal value, Optimal solution  Encodings  //Data Structure.
Week 10Complexity of Algorithms1 Hard Computational Problems Some computational problems are hard Despite a numerous attempts we do not know any efficient.
Complexity 25-1 Complexity Andrei Bulatov Counting Problems.
CSC401 – Analysis of Algorithms Chapter 13 NP-Completeness Objectives: Introduce the definitions of P and NP problems Introduce the definitions of NP-hard.
CSCI 3160 Design and Analysis of Algorithms Tutorial 10 Chengyu Lin.
1 The Theory of NP-Completeness 2 Cook ’ s Theorem (1971) Prof. Cook Toronto U. Receiving Turing Award (1982) Discussing difficult problems: worst case.
NP-Complete Problems. Running Time v.s. Input Size Concern with problems whose complexity may be described by exponential functions. Tractable problems.
1 Chapter 34: NP-Completeness. 2 About this Tutorial What is NP ? How to check if a problem is in NP ? Cook-Levin Theorem Showing one of the most difficult.
NP-complete Problems Prof. Sin-Min Lee Department of Computer Science.
CSE 589 Part V One of the symptoms of an approaching nervous breakdown is the belief that one’s work is terribly important. Bertrand Russell.
CS6045: Advanced Algorithms NP Completeness. NP-Completeness Some problems are intractable: as they grow large, we are unable to solve them in reasonable.
Lecture. Today Problem set 9 out (due next Thursday) Topics: –Complexity Theory –Optimization versus Decision Problems –P and NP –Efficient Verification.
CSCI 2670 Introduction to Theory of Computing December 2, 2004.
Spring 2007NP-Completeness1 x 1 x 3 x 2 x 1 x 4 x 3 x 2 x
1 1. Which of these sequences correspond to Hamilton cycles in the graph? (a) (b) (c) (d) (e)
The NP class. NP-completeness Lecture2. The NP-class The NP class is a class that contains all the problems that can be decided by a Non-Deterministic.
CSE 332: NP Completeness, Part II Richard Anderson Spring 2016.
ICS 353: Design and Analysis of Algorithms NP-Complete Problems King Fahd University of Petroleum & Minerals Information & Computer Science Department.
NP-Completeness A problem is NP-complete if: It is in NP
The NP class. NP-completeness
NP-Completeness (2) NP-Completeness Graphs 4/13/2018 5:22 AM x x x x x
Data Structures and Algorithm Analysis Lecture 24
Graphs 4/13/2018 5:25 AM Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 NP-Completeness.
P & NP.
Chapter 10 NP-Complete Problems.
Richard Anderson Lectures NP-Completeness
Richard Anderson Lecture 26 NP-Completeness
Great Theoretical Ideas in Computer Science
NP-Completeness (2) NP-Completeness Graphs 7/23/ :02 PM x x x x
NP-Completeness (2) NP-Completeness Graphs 7/23/ :02 PM x x x x
Hard Problems Introduction to NP
NP-Completeness Yin Tat Lee
CS21 Decidability and Tractability
Intro to Theory of Computation
NP-Completeness Merge Sort 11/16/2018 2:30 AM Spring 2007
NP-Completeness NP-Completeness Graphs 11/16/2018 2:32 AM x x x x x x
Analysis and design of algorithm
ICS 353: Design and Analysis of Algorithms
NP-Completeness (2) NP-Completeness Graphs 11/23/2018 2:12 PM x x x x
Richard Anderson Lecture 25 NP-Completeness
NP-Completeness NP-Completeness Graphs 12/3/2018 2:46 AM x x x x x x x
NP-Completeness NP-Completeness Graphs 12/6/2018 1:58 PM x x x x x x x
Chapter 34: NP-Completeness
1. for (i=0; i < n; i+=2) if (A[i] > A[i+1]) swap(A[i], A[i+1])
Richard Anderson Lecture 30 NP-Completeness
Chapter 11 Limitations of Algorithm Power
NP-Complete Problems.
Richard Anderson Lecture 27 NP Completeness
NP-completeness The Chinese University of Hong Kong Fall 2008
NP-Completeness Yin Tat Lee
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979.
P, NP and NP-Complete Problems
P, NP and NP-Complete Problems
Richard Anderson Lecture 25 NP Completeness
Our old list of problems
RAIK 283 Data Structures & Algorithms
NP-Completeness (2) NP-Completeness Graphs 7/9/2019 6:12 AM x x x x x
Presentation transcript:

NP-Completeness NP-Completeness Graphs 5/7/2018 11:49 PM x x x x x x x 3 3 4 4 12 22 32 11 13 21 23 31 33 NP-Completeness

Outline and Reading P and NP (§13.1) NP-completeness (§13.2) Definition of P Definition of NP Alternate definition of NP NP-completeness (§13.2) Definition of NP-hard and NP-complete The Cook-Levin Theorem NP-Completeness

Running Time Revisited Input size, n To be exact, let n denote the number of bits in a nonunary encoding of the input All the polynomial-time algorithms studied so far in this course run in polynomial time using this definition of input size. Exception: any pseudo-polynomial time algorithm ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142 NP-Completeness

Dealing with Hard Problems What to do when we find a problem that looks hard… I couldn’t find a polynomial-time algorithm; I guess I’m too dumb. NP-Completeness (cartoon inspired by [Garey-Johnson, 79])

Dealing with Hard Problems Sometimes we can prove a strong lower bound… (but not usually) I couldn’t find a polynomial-time algorithm, because no such algorithm exists! NP-Completeness (cartoon inspired by [Garey-Johnson, 79])

Dealing with Hard Problems NP-completeness let’s us show collectively that a problem is hard. I couldn’t find a polynomial-time algorithm, but neither could all these other smart people. NP-Completeness (cartoon inspired by [Garey-Johnson, 79])

Polynomial-Time Decision Problems To simplify the notion of “hardness,” we will focus on the following: Polynomial-time as the cut-off for efficiency Decision problems: output is 1 or 0 (“yes” or “no”) Examples: Does a given graph G have an Euler tour? Does a text T contain a pattern P? Does an instance of 0/1 Knapsack have a solution with benefit at least K? Does a graph G have an MST with weight at most K? NP-Completeness

Problems and Languages A language L is a set of strings defined over some alphabet Σ Every decision algorithm A defines a language L L is the set consisting of every string x such that A outputs “yes” on input x. We say “A accepts x’’ in this case Example: If A determines whether or not a given graph G has an Euler tour, then the language L for A is all graphs with Euler tours. NP-Completeness

The Complexity Class P A complexity class is a collection of languages P is the complexity class consisting of all languages that are accepted by polynomial-time algorithms For each language L in P there is a polynomial-time decision algorithm A for L. If n=|x|, for x in L, then A runs in p(n) time on input x. The function p(n) is some polynomial NP-Completeness

The Complexity Class NP We say that an algorithm is non-deterministic if it uses the following operation: Choose(b): chooses a bit b Can be used to choose an entire string y (with |y| choices) We say that a non-deterministic algorithm A accepts a string x if there exists some sequence of choose operations that causes A to output “yes” on input x. NP is the complexity class consisting of all languages accepted by polynomial-time non-deterministic algorithms. NP-Completeness

NP example Problem: Decide if a graph has an MST of weight K Algorithm: Non-deterministically choose a set T of n-1 edges Test that T forms a spanning tree Test that T has weight at most K Analysis: Testing takes O(n+m) time, so this algorithm runs in polynomial time. NP-Completeness

The Complexity Class NP Alternate Definition We say that an algorithm B verfies the acceptance of a language L if and only if, for any x in L, there exists a certificate y such that B outputs “yes” on input (x,y). NP is the complexity class consisting of all languages verified by polynomial-time algorithms. We know: P is a subset of NP. Major open question: P=NP? Most researchers believe that P and NP are different. NP-Completeness

NP example (2) Problem: Decide if a graph has an MST of weight K Verification Algorithm: Use as a certificate, y, a set T of n-1 edges Test that T forms a spanning tree Test that T has weight at most K Analysis: Verification takes O(n+m) time, so this algorithm runs in polynomial time. NP-Completeness

Equivalence of the Two Definitions Suppose A is a non-deterministic algorithm Let y be a certificate consisting of all the outcomes of the choose steps that A uses We can create a verification algorithm that uses y instead of A’s choose steps If A accepts on x, then there is a certificate y that allows us to verify this (namely, the choose steps A made) If A runs in polynomial-time, so does this verification algorithm Suppose B is a verification algorithm Non-deterministically choose a certificate y Run B on y If B runs in polynomial-time, so does this non-deterministic algorithm NP-Completeness

An Interesting Problem A Boolean circuit is a circuit of AND, OR, and NOT gates; the CIRCUIT-SAT problem is to determine if there is an assignment of 0’s and 1’s to a circuit’s inputs so that the circuit outputs 1. NP-Completeness

CIRCUIT-SAT is in NP Non-deterministically choose a set of inputs and the outcome of every gate, then test each gate’s I/O. NP-Completeness

NP-Completeness A problem (language) L is NP-hard if every problem in NP can be reduced to L in polynomial time. That is, for each language M in NP, we can take an input x for M, transform it in polynomial time to an input x’ for L such that x is in M if and only if x’ is in L. L is NP-complete if it’s in NP and is NP-hard. NP poly-time L NP-Completeness

Cook-Levin Theorem CIRCUIT-SAT is NP-complete. We already showed it is in NP. To prove it is NP-hard, we have to show that every language in NP can be reduced to it. Let M be in NP, and let x be an input for M. Let y be a certificate that allows us to verify membership in M in polynomial time, p(n), by some algorithm D. Let S be a circuit of size at most O(p(n)2) that simulates a computer (details omitted…) NP poly-time M CIRCUIT-SAT NP-Completeness

Cook-Levin Proof We can build a circuit that simulates the verification of x’s membership in M using y. S Let W be the working storage for D (including registers, such as program counter); let D be given in RAM “machine code.” Simulate p(n) steps of D by replicating circuit S for each step of D. Only input: y. Circuit is satisfiable if and only if x is accepted by D with some certificate y Total size is still polynomial: O(p(n)3). D D D < p(n) W W W cells Output 0/1 from D S S Inputs y y y n x x x p(n) steps NP-Completeness

Some Thoughts about P and NP CIRCUIT-SAT NP-complete problems live here Some Thoughts about P and NP Belief: P is a proper subset of NP. Implication: the NP-complete problems are the hardest in NP. Why: Because if we could solve an NP-complete problem in polynomial time, we could solve every problem in NP in polynomial time. That is, if an NP-complete problem is solvable in polynomial time, then P=NP. Since so many people have attempted without success to find polynomial-time solutions to NP-complete problems, showing your problem is NP-complete is equivalent to showing that a lot of smart people have worked on your problem and found no polynomial-time algorithm. NP-Completeness